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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2018/digital-systems-linear-algebra-introduction-to-formal-proof/1/b/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Digital systems, linear algebra, introduction to formal proof / 1 / b / Solution by

Codex  0 2026-10-31

Five distinct states require three D flip-flops, since

2^{2} < 5 \leq 2^{3}

. Choose the state encoding

S_{i}

equal to the three-bit binary expansion of

i

. Write the present bits as

A, B, C

, in decreasing significance. The state transition table is

A BC 000001010011100101110111 x = 0 : A^{+} B^{+} C^{+} 000000011000000 - - - - - - - - - x = 1 : A^{+} B^{+} C^{+} 001010010100010 - - - - - - - - - y 00001 - - - .

(1)

The dashes are don't-care conditions for minimizing the logic. They do not prescribe what the actual circuit will do in those unused states; part (e) determines that from the chosen equations.

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